Japanese professional ranks go from 1 dan to 9 dan, with 9 dan being strongest. In theory, by virtue of their superior training and rigorous discipline, professional players are always stronger than amateurs. In practice, a few ultra-strong amateurs 7 dans have occasionally beaten a few of the weaker professional 1 dans in even games. (That's what I heard from the folks who taught me the game; I have since learned that strong amateurs can sometimes beat strong pros.)
BillSpight: I read years ago (something by Sakata) that top amateurs could match pro 5th or 6th dans. Recently I have read claims such as the one above. Not that it matters, but I doubt it. :-)
TakeNGive: Particularly in light of the 2001 San Francisco Oza results, I have to say that very strong amateurs are as strong as professionals.
The steps between pro ranks are said to be smaller than between amateur ranks -- about 1/3 of a stone.
BillSpight: The traditional pro handicapping made a difference of three ranks per stone. However, with the new (post-WWII) pro rankings, there seems to be a two-stone difference between 9-dans and shodans, which translates to about 1/4 stone per rank.
exswoo: So, if one were to convert the pro ratings back to amateur ratings, a 9p would be more or less a 10d amateur? That's interesting..., and since the oft-repeated saying is that a grandmaster would need a 3-stone handicap to play against God, God must be 22p :)
Andre Engels: In Europe, a 7D European rank is considered about equivalent in strength to a 1-2P professional.
BlueWyvern: From a conversation I had with a top European amateur, currently new Japanese pro's have the strength about of a 5 dan pro. Apparently the new crop has been getting tougher lately, so for someone of 1 dan strength, you need to take a pro who has been 1 dan for 10 or more years.
Calvin: Some data from the Pro-Am Honinbo Match might be interesting, although I can't find it compiled anywhere. Usually white gives komi and handicap. Here's a link where Iwai Ryuichi won against O Meien playing black with 2 stones and -5 komi. The result was B+4, so that's a two-stone game:
http://www.msoworld.com/mindzine/news/orient/go/sgf/honproam38.sgf
Harada Minoru, a very strong Japanese amateur, took three stones and -2.5 komi against Cho Chikun in the 1997 Pro-Am Honinbo and lost.
Bob McGuigan: It's interesting to see some of Harada's other results: 1963 and 64 defeated Sakata with 2 stones, 1968 defeated Rin with 3.5 reverse komi, 1979 jigo with Cho Chikun with 5 point reverse komi. The results of all these matches are in the Japanese yearbooks.
Dave: The komi system used is explained on Amateur Honinbo vs Pro Honinbo 2002.